Astronomy
&
Astrophysics
A&A 539, A30 (2012)
DOI: 10.1051/0004-6361/201118268
c ESO 2012
Numerical calculation of convection with reduced
speed of sound technique
(Research Note)
H. Hotta1 , M. Rempel2 , T. Yokoyama1 , Y. Iida1 , and Y. Fan2
1
2
Department of Earth and Planetary Science, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, 113-0033 Tokyo, Japan
e-mail: [email protected]
High Altitude Observatory, National Center for Atmospheric Research, Boulder, CO, USA
Received 13 October 2011 / Accepted 31 December 2011
ABSTRACT
Context. The anelastic approximation is often adopted in numerical calculations with low Mach numbers, such as those including
stellar internal convection. This approximation requires so-called frequent global communication, because of an elliptic partial differential equation. Frequent global communication is, however, negative factor for the parallel computing performed with a large number
of CPUs.
Aims. We test the validity of a method that artificially reduces the speed of sound for the compressible fluid equations in the context
of stellar internal convection. This reduction in the speed of sound leads to longer time steps despite the low Mach number, while the
numerical scheme remains fully explicit and the mathematical system is hyperbolic, thus does not require frequent global communication.
Methods. Two- and three-dimensional compressible hydrodynamic equations are solved numerically. Some statistical quantities of
solutions computed with different effective Mach numbers (owing to the reduction in the speed of sound) are compared to test the
validity of our approach.
Results. Numerical simulations with artificially reduced speed of sound are a valid approach as long as the effective Mach number
(based on the lower speed of sound) remains less than 0.7.
Key words. methods: numerical – stars: interiors – convection – Sun: interior
1. Introduction
Turbulent thermal convection in the solar convection zone plays
a key role for the maintenance of large scale flows (differential
rotation, meridional flow) and solar magnetic activity. The angular momentum transport of convection maintains the global
mean flows. Global flows relate to the generation of the global
magnetic field, i.e. the solar dynamo. Differential rotation bends
the pre-existing poloidal field and generates the strong toroidal
field (the Ω effect), and mean meridional flow transports the
magnetic flux equatorward at the base of the convection zone
(Choudhuri et al. 1995; Dikpati & Charbonneau 1999). The internal structures of the solar differential rotation and the meridional flow are revealed by the helioseismology (see review by
Thompson et al. 2003). Some mean field studies have reproduced these global flows (Kichatinov & Rüdiger 1993; Küker
& Stix 2001; Rempel 2005; Hotta & Yokoyama 2011). These
studies, however, used some kinds of parameterization of the
turbulent convection, i.e. turbulent viscosity and turbulent angular momentum transport. Thus, a self-consistent thorough understanding of global structure requires the detailed investigation
of the turbulent thermal convection. Turbulent convection is also
important when describing the magnetic field itself. The strength
of the next solar maximum in the predictions of the mean field
model depends significantly on the turbulent diffusivity (Dikpati
& Gilman 2006; Choudhuri et al. 2007; Yeates et al. 2008). In
addition, the turbulent diffusion has an important influence on
the parity of solar global field, the strength of polar field and so
on (Hotta & Yokoyama 2010a,b).
There have been numerous LES numerical simulations of
the solar and stellar convection (Gilman 1977; Gilman & Miller
1981; Glatzmaier 1984; Miesch et al. 2000, 2006; Brown et al.
2008) and their magnetic fields (Gilman & Miller 1981; Brun
et al. 2004; Brown et al. 2010, 2011). In these studies, the anelastic approximation is adopted to avoid any of the difficulties
caused by the high speed of sound. At the base of the convection zone, the speed of sound is about 200 km s−1 . In contrast,
the speed of convection is thought to be 50 m s−1 (Stix 2004).
The time step must therefore be shorter owing to the CFL condition in an explicit fully compressible method, even when we are
interested in phenomena related to convection. In the anelastic
approximation, the equation of continuity is treated as
∇ · (ρ0 u) = 0,
(1)
where ρ0 is the stratified background density and u denotes the
velocity. The anelastic approximation assumes that the speed of
sound is essentially infinite, resulting in an instantaneous adjustment of pressure to flow changes. This is achieved by solving
an elliptic equation for the pressure, which filters out the propagation of sound waves. As a result, the time step is only limited by the much lower flow velocity. However, owing to the
existence of an elliptic the anelastic approximation has a weak
point. When performing the numerical calculation using parallel
Article published by EDP Sciences
A30, page 1 of 7
A&A 539, A30 (2012)
computing, we must employ the frequent global communication.
At the present time, the efficiency of scaling in parallel computing is saturated with about 2000–3000 CPUs in solar global simulations with the pseudo-spectral method (Miesch, priv. comm.).
Higher resolution, however, is needed to understand the precise
mechanism of the angular momentum and energy transport by
the turbulent convection and the behavior of magnetic field especially in thin magnetic flux tube.
In this paper, we test the validity of a different approach to
circumvent the severe numerical time step constraints in low
Mach number flows. We use a method in which the speed of
sound is reduced artificially by transforming the equation of continuity to (see, e.g., Rempel 2005)
∂ρ
1
= − 2 ∇ · (ρu),
∂t
ξ
(2)
where t denotes the time. Using this equation, the effective speed
of sound becomes ξ times smaller, but the dispersion relationship
for sound waves remains otherwise unchanged (wave speed decreases equally for all wavelengths). Since this technique does
not change the hyperbolic character of the underlying equations,
the numerical treatment can remain fully explicit, thus does not
require global communication in parallel computing. We refer
to the technique as the Reduced Speed of Sound Technique
(RSST). This technique was used previously by Rempel (2005,
2006) in mean field models of solar differential rotation and nonkinematic dynamos, which essentially solve the full set of timedependent axisymmetric MHD equations. Those solutions were
however restricted to the relaxation toward a stationary state or
very slowly varying problems on the timescale of the solar cycle. Here we apply this approach to thermal convection, where
the intrinsic timescales are substantially shorter. To this end, we
study two- and three-dimensional (3D) convection, in particular
the latter will be non-stationary and turbulent.
The detailed setting of test calculation is given in Sect. 2. The
results of our calculations are given in Sect. 3. We summarize
our paper and give discussion of the RSST in Sect. 4.
2. Model
In this section, we describe the detailed settting of this study.
2.1. Equations
The two- or three-dimensional equation of continuity, equation
of motion, equation of energy, equation of state, are solved in
Cartesian coordinate (x, z) or (x, y, z), where x and y denote the
horizontal direction and z denotes the vertical direction. The basic assumptions underlying this study are as:
1. the time-independent hydrostatic reference state;
2. that the perturbations caused by thermal convection are
small, i.e. ρ1 ρ0 and p1 p0 , here ρ0 and p0 denote
the reference state values, whereas ρ1 and p1 are the fluctuations of density and pressure, respectively. Thus a linearized
equation of state is used as Eq. (6);
3. that the profile of the reference entropy s0 (z) is a steady state
solution of the thermal diffusion equation ∇·(Kρ0 T 0 ∇s0 ) = 0
with constant K.
A30, page 2 of 7
The formulations are almost identical to those of Fan et al.
(2003), and given by
1
∂ρ1
= − 2 ∇ · (ρ0 u),
∂t
ξ
∇p1 ρ1
∂u
1
= −(u · ∇)u −
− gez + ∇ · Π,
∂t
ρ0
ρ0
ρ0
∂s1
1
= −(u · ∇)(s0 + s1 ) +
∇ · (Kρ0 T 0 ∇s1 )
∂t
ρ0 T 0
γ−1
(Π · ∇) · u,
+
p0
ρ1
p1 = p0 γ + s1 ,
ρ0
(3)
(4)
(5)
(6)
where T 0 (z), and s0 (z) denote the reference temperature and entropy, respectively, and ez denotes the unit vector along the zdirection, γ is the ratio of specific heats, with the value for an
ideal gas being γ = 5/3, and s1 denotes the fluctuation of entropy
from reference atmosphere. We note that the entropy is normalized by a value of the specific heat capacity at constant volume
cv . The quantity g is the gravitational acceleration, which is assumed to be constant. The quantity Π denotes the viscous stress
tensor
∂vi ∂v j 2
Πi j = ρ0 ν
+
− (∇ · u)δi j ,
(7)
∂x j ∂xi 3
and ν and K denote the viscosity and thermal diffusivity, respectively, and ν and K are assumed to be constant throughout the
simulation domain. We assume for the reference atmosphere a
weakly superadiabatically stratified polytrope
m
z
ρ0 (z) = ρr 1 −
,
(8)
(m + 1)Hr
m+1
z
p0 (z) = pr 1 −
,
(9)
(m + 1)Hr
z
T 0 (z) = T r 1 −
,
(10)
(m + 1)Hr
p0
,
(11)
Hp (z) =
ρ0 g
ds0
γδ(z)
=−
,
(12)
dz
Hp (z)
ρr
,
(13)
δ(z) = δr
ρ0 (z)
where ρr , pr , T r , Hr , and δr denote the values of ρ0 , p0 , T 0 ,
H0 (the pressure scale height), and δ (the non-dimensional superadiabaticity) at the bottom boundary z = 0. Since |δ| 1,
the value m is nearly equal to the adiabatic value, meaning
that m = 1/(γ − 1). The strength of the diffusive parameter
ν and K is expressed in terms of the non-dimensional parameters: the Reynolds number Re ≡ vc Hr /ν and the Prandtl number
Pr ≡ ν/K, where the velocity unit vc ≡ (8δr gHr )1/2 . We note that
in this paper the unit of time is Hr /vc . In all calculations, we set
Pr = 1.
2.2. Boundary conditions and numerical method
We solve Eqs. (3)–(6) numerically. At the horizontal boundaries
(x = 0, L x and y = 0, Ly ), periodic boundary conditions are
adopted for all variables. At the top and the bottom boundaries,
H. Hotta et al.: Convection with RSST (RN)
Table 1. Parameters for numerical simulations.
Case
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Dimension
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3
3
L x × Ly × Lz (Hr )
N x × Ny × Nz
8.72 × 2.18
8.72 × 2.18
8.72 × 2.18
8.72 × 2.18
8.72 × 2.18
8.72 × 8.72 × 2.18
8.72 × 8.72 × 2.18
8.72 × 8.72 × 2.18
8.72 × 8.72 × 2.18
8.72 × 8.72 × 2.18
8.72 × 8.72 × 2.18
8.72 × 8.72 × 2.18
8.72 × 8.72 × 2.18
8.72 × 8.72 × 2.18
8.72 × 8.72 × 2.18
8.72 × 8.72 × 2.18
8.72 × 8.72 × 2.18
8.72 × 8.72 × 2.18
δr
−6
384 × 96
384 × 96
384 × 96
384 × 96
384 × 96
384 × 384 × 96
384 × 384 × 96
384 × 384 × 96
384 × 384 × 96
384 × 384 × 96
384 × 384 × 96
384 × 384 × 96
384 × 384 × 96
384 × 384 × 96
384 × 384 × 96
384 × 384 × 96
384 × 384 × 96
384 × 384 × 96
1 × 10
1 × 10−6
1 × 10−6
1 × 10−6
1 × 10−6
1 × 10−4
1 × 10−4
1 × 10−4
1 × 10−4
1 × 10−4
1 × 10−4
1 × 10−4
1 × 10−4
1 × 10−3
1 × 10−3
1 × 10−3
1 × 10−3
1 × 10−3
Re
ξ
260
260
260
260
260
300
300
300
300
300
300
300
300
300
300
300
300
300
1
10
30
50
80
1
5
10
15
20
40
80
20/(δ/δr )1/2
1
5
10
15
20
Notes. In case 13, the value of ξ is 20 at the bottom and 4.5 at the top boundary.
impenetrative and stress-free boundary conditions are adopted
for the velocities and the entropy is fixed to
vz = 0,
∂v x
= 0,
∂z
∂vy
= 0,
∂z
s1 = 0.
(14)
(15)
(16)
(17)
The longest allowed time step, Δtmax , is determined by the
CFL criterion. When both advection and diffusion terms are included in calculation, the time step reads
Δtmax = min(Δtad , Δtv ),
(24)
where
Δtad = cad
min(Δx, Δy, Δz)
,
ctot
(25)
and ctot is the total wave speed
At both the top and bottom boundaries (z = 0 and z = Lz ), we
set p1 in the ghost cells such that the right hand side of the zcomponent of Eq. (4) is zero at the boundary (which is between
ghost cells and domain cells), where the ghost cells are the cells
beyond the physical boundary.
We adopt the fourth-order space-centered difference for each
derivative. The first spatial derivatives of quantity q is given by
∂q
1
(−qi+2 + 8qi+1 − 8qi−1 + qi−2 ),
=
(18)
∂x i 12Δx
ctot = |v| + cs ,
where i denotes the index of the grid position along a particular spatial direction. The numerical solution of the system
is advanced in time with an explicit fourth-order Runge-Kutta
scheme. The system of partial equations can be written as
and cad and cv are safety factors of order unity.
Using δr = 1 × 10−4 , the original speed of sound is about
35vc at the bottom and 12vc at the top boundary, respectively. In
all calculations, Δx ∼ 2.3 × 10−2 Hr . Thus, if we use cad = 1 and
cv = 1, Δtad = 6.4 × 10−4 Hr /vc and Δtv = 1.5 × 10−1 Hr /vc . For
all the ξ values considered in this paper, the time step remains
restricted by the (reduced) speed of sound, thus the calculation
is about ξ times more efficient with RSST.
∂U
= R(U)
∂t
(19)
for Un+1 , which is the value at tn+1 = (n + 1)Δt is calculated in
four steps
Δt
R(Un ),
4
Δt
= Un + R(Un+ 1 ),
4
3
Δt
= Un + R(Un+ 1 ),
3
2
= Un + ΔtR(Un+ 1 ).
Un+ 1 = Un +
4
Un+ 1
3
Un+ 1
2
Un+1
2
(20)
(21)
(22)
(23)
(26)
where the effective speed of sound is expressed as
1
p0
γ ·
cs =
ξ
ρ0
(27)
The time step determined by the diffusion term is
Δtv = cv
min(Δx2 , Δy2 , Δz2 )
,
max(K, ν)
(28)
3. Results
3.1. Two-dimensional study
We carried out four two-dimensional (2D) calculations with the
RSST and one calculation without approximation (case 1–5).
The values of the free parameters are given in Table 1. The superadiabaticity (δ) is 1 × 10−6 at the bottom and about 2 × 10−5
at the top boundary. It is almost the same superadiabaticity value
as at the base of the solar convection zone. Figure 1 shows the
A30, page 3 of 7
A&A 539, A30 (2012)
Entropy (δr cv)
(a) Maximum Mach number
= 0.50
2.1686
2.0
z (Hr)
1.5
0.1000
10
1.0
0.100
-10
0.0
0
ξ=1
ξ=10
ξ=20
ξ=40
ξ=80
0.0100
0
0.5
(b) Mach number
1.000
20
0.010
0.0010
0.0114
0.0
4
6
8
0.001
0
x (Hr)
Entropy (δr cv)
2.1686
2.0
200
300
time (Hr/vc)
400
(c) Horizontal RMS velocity
20
1.0
10
0.8
0
0.6
0.5
1.0
1.5
z (Hr)
2.0
(d) Vertical RMS velocity
(e) Relative variation
0.4
-10
0.0
0
t=400.00
0.3
vz (vc)
1.0
0.5
0.4
0.0
8.7
4
6
0.2
8
0.0
0.0
Fig. 1. Time-development of entropy in a 2D calculation with parameters of case 1. (Top panel) t = 20. (Bottom panel) t = 400.
time-development of entropy. In the beginning, non-linear timedependent convection occurred (top panel), which changed to a
steady state at later times (bottom panel). For the steady state, we
compared the root mean squared (rms) velocity with different ξ.
The rms velocity was defined as
Lx Ly
1
vrms =
v2 dxdy.
(29)
L x Ly 0
0
Figure 2 shows our results where the value of ξ is from 1 to 80.
The effective Mach number is defined as MA = vrms /cs . If we
have larger ξ value than 80, we cannot reach a stationary state,
since there are some shocks generated by supersonic convection.
The discussion of unsteady convection is given in the next session of 3D calculations. Even though the Mach number reaches
0.6 using ξ = 80 (panel a), the horizontal and vertical rms velocities are almost identical to those calculated with ξ = 1 (without
RSST). The ratio of the rms velocities for each ξ and ξ = 1 are
shown in Fig. 2e. The difference is always a few percent. This
result is unsurprising since in the stationary state the equation of
continuity becomes
1
∇ · (ρ0 u),
ξ2
(30)
whose solution must not depend on the value of ξ. We note that
the cell size is to some degree affected by the aspect ratio of the
domain. This does not affect our conclusions since this influence
is the same for all considered values of ξ. To confirm the robustness of our conclusions, we repeated this experiment with a
wider domain (26.16Hr × 2.18Hr instead of 8.72Hr × 2.18Hr ) in
which we find four steady convection cells with about 10% different rms velocities. In addition we find a dependence on ξ that
is similar to that shown in Fig. 2, i.e. the solutions show differences of only a few percent as long as ξ < 80. In this section, we
confirm that the RSST is valid for the 2D stationary convection
when the effective Mach number is less than 1, i.e. ξ < 80 with
δr = 1 × 10−6 . If we use δr = 1 × 10−4 (result is not shown), the
criterion becomes ξ < 8 for the 2D calculation.
As a next step, we investigate the dependence of the linear
growth rate on ξ during the initial (time-dependent) relaxation
phase toward the final stationary state. Figure 3 shows the linear growth of maximum perturbation density ρ1 with different
ξ. Black and red lines show the results with δr = 1 × 10−6 and
δr = 1 × 10−4 , respectively. These calculation parameters are
not given in Table 1. In the calculation with δ = 1 × 10−6 , the
growth rate decreases for values of ξ > 300, whereas it occurs
at ξ > 30 in the calculation with δr = 1 × 10−4 . The reason
can be explained as follows. In the convective instability, upflow
A30, page 4 of 7
1.00
0.98
0.96
x (Hr)
0.5
1.0
1.5
z (Hr)
2.0
0.0
0.0
0.5
1.0
1.5
z (Hr)
2.0
0.0
0.5
1.0
1.5
z (Hr)
2.0
Fig. 2. Some quantities in 2D calculation. a) Maximum Mach number of
each time step. b) Distribution of Mach number estimated with the rms
velocity. c) Distribution of horizontal rms velocity vh . d) Distribution of
vertical rms velocity. e) Ratio of the rms velocities for each ξ and ξ = 1.
2.0
growth rate (vc/Hr)
2
1.02
0.2
0.1
0.0114
0=
0.0001
0.0
1.04
1.5
z (Hr)
100
= 0.50
vh (vc)
t=20.00
8.7
2
Dependence of growth rate on ξ
1.8
1.6
1.4
1.2
1.0
1
10
ξ
100
1000
Fig. 3. Behavior in linear phase. Dependence of growth rate on ξ is
shown. Black and red lines show the results with δr = 1 × 10−6 and
1 × 10−4 , respectively.
(downflow) generates a positive (negative) entropy perturbation
and then a negative (positive) density perturbation is generated
by the sound wave. If the speed of sound is fairly slow, the generation mechanism of density perturbation is ineffective. In the
calculation with δr = 1 × 10−6 (δr = 1 × 10−4 ), the growth rate
with ξ = 200 (ξ = 20), however, is almost same as that with
ξ = 1, even though flow with ξ = 200 (ξ = 20) is expected to be
the Mach number Ma = 1.3, i.e. supersonic convection flow in
the saturated state.
3.2. Three-dimensional study
We now investigate the suitability of the RSST for 3D unsteady
thermal convections (case 6–13). The value of superadiabaticity at the bottom boundary is 1 × 10−4 and at the top 2 × 10−3 .
Although this value is relatively large compared to solar value,
the expected speed of convection is much smaller than the speed
of sound, hence is small enough to allow us to investigate the
validity of the RSST. The entropy of 3D convections with ξ = 1,
20, and 80 at t = 100Hr /vc are shown in Fig. 4. The convection
is completely unsteady and turbulent (animation is provided).
The appearance of convection with ξ = 20 is almost the same as
that with ξ = 1. We verify this using the Fourier transformation
and auto-detection technique. However, the appearance of convection with ξ = 80 (bottom panel) differs completely from the
H. Hotta et al.: Convection with RSST (RN)
-5
2
4
x(Hr)
6
0
-1
2
s1 (cvδ) at z=2.18Hr
4
0
2
-5
2
0
0
-10
0
0
4
x(Hr)
6
y(Hr)
y(Hr)
5
2
8
4
0
4
0
2
-5
2
0
0
-10
0
0
6
y(Hr)
y(Hr)
6
4
x(Hr)
(a) Mach number
0.10
2
0.015
1.0 1.5
Height (Hr)
8
4
x(Hr)
6
0
4
x(Hr)
6
0.4
0.4
0.3
0.3
0.2
0.2
0.015
0.005
0.005
0.000
0.000
-0.005
-0.005
2.0
#11
#15
#12
8
0.1
0.0
1.0 1.5
Height (Hr)
2.0
0.5
1.0 1.5
Height (Hr)
2.0
(e) <-ρ0v⋅(v⋅∇)v>
ξ=1
ξ=5
ξ=10
ξ=15
ξ=20
ξ=40
ξ=80
ξ=20/(δ/δr)1/2
0.5
1.0
1.5
Height (Hr)
#10
#09
4
#08
#07
#06
#01
2
#03
#02
0
2
4
x (Hr)
6
8
0
2
4
x (Hr)
6
8
Fig. 7. Detection of convective cell. a) Original contour of density perturbation at z = 2.18Hr . b) Distribution of detected convection cell.
Color and label (#n) show each convective cell.
-0.010
1.0
1.5
Height (Hr)
#13
6
4
0
0
(c) Horizontal RMS velocity
0.5
100
(b) Detection of convective cell
#17
-2
2
10
k = (k2x +k2y )1/2
#16
8
2
-1
0.6
0.5
10-8
1
100
#05
4
(b) Vertical RMS velocity
0.010
0.5
10
k = (k2x +k2y )1/2
#04
1
0.5
0.010
-0.010
100
10-6
6
8
6
0.6
2.0
(d) <-v⋅(∇ p1+ρ1g)>
10
k = (k2x +k2y )1/2
10-4
#14
2
0.0
0.5
10-8
1
(a) ρ1 at z=2.18Hr
-2
0.1
0.01
10-8
1
8
Fig. 4. Snapshots of entropy of the 3D convection. Top, middle, and
bottom panels correspond to ξ = 1, 10, and 80, respectively. Left (right)
panels show entropy at top (bottom) boundary. (Animation is provided,
and the difference between ξ = 1 and 80 is most clearly visible in the
animation of Fig. 4, which is provided with the online version.)
1.00
10-6
-1
8
5
2
10-4
10-6
ξ=1
ξ=5
ξ=10
ξ=15
ξ=20
ξ=20/(δ/δr)1/2
10-2
Fig. 6. Comparison of horizontal velocity spectra with different ξ. a)
z = 2.18Hr b) z = 1.09Hr c) z = 0. Velocities are averaged in time
between t = 100 and 200Hr /vc .
2
1
10-2
s1 (cvδ) at z=0
10
ξ=80
8
6
s1 (cvδ) at z=2.18Hr
8
6
8
6
ξ=20
4
x(Hr)
s1 (cvδ) at z=0
10
8
10-4
-2
0
0
8
10-2
100
amplitude (vc)
4
y (Hr)
2
1
2
-10
ξ=1
6
(c) z=0
100
y (Hr)
4
0
2
amplitude (vc)
5
(b) z=1.09Hr
100
8
y(Hr)
y(Hr)
6
0
0
(a) z=2.18Hr
s1 (cvδ) at z=0
10
amplitude (vc)
s1 (cvδ) at z=2.18Hr
8
2.0
Fig. 5. Some quantities averaged in time between t = 100 and 200Hr /vc
in 3D calculations. a) Distribution of Mach number estimated with the
rms velocity. b) Distribution of vertical rms velocity vh . c) Distribution
of horizontal rms velocity. d) Distribution of rms power density of pressure and buoyancy. e) Distribution of rms power density of inertia.
others. This difference is most clearly visible in the animation of
Fig. 4 that is provided with the online version.
We estimate the rms velocities for different ξ by calculating the average of values between t = 100 to 200Hr /vc (Fig. 5:
panel b and c). Without the RSST, i.e. ξ = 1 and using δr =
1×10−4 , the Mach number is 1×10−2 at the bottom and 4×10−2 at
the top boundary. The rms velocities at ξ = 40 and 80 differ from
those at the ξ = 1 by less than 15% and 30%, respectively. When
we adopt ξ = 40 and 80, the Mach number estimated by rms velocity exceeds unity (Fig. 5: panel a), i.e. supersonic convection.
This supersonic downflow frequently generates shocks and positive entropy perturbation, thus downflow is decelerated. This
is the reason why the rms velocities with large ξ(=40, 80) are
small. When ξ = 5, 10, 15, and 20, however, the rms velocities
are in good agreement with that for ξ = 1. We estimate the rms
power density of pressure, buoyancy, and inertia (Fig. 5). This
results display almost the same trend as for the rms velocities.
The power profiles ξ = 5, 10, 15, 20 agree with each other and
those with ξ = 40, 80 disagree with those with ξ = 1. These results show that the RSST is a valid technique at least for ξ = 20,
at which the Mach number is around 0.7. We note that the convection pattern among our 2D and 3D cases varies substantially.
As a consequence, the ξ values for which the validity of RSST
breaks down are also different in the 2D and 3D setups.
In Fig. 6, we compare the average spectral amplitudes for
different values of ξ. There is no significant difference between
the spectral amplitudes for different values of ξ in the range
from 1 to 20.
We investigate the distribution of cell sizes at the top boundary. This value is strongly related to the turbulent diffusivity and
transport of either angular momentum or energy. The method for
detecting the cell is as follows. At the boundary of the cell i.e.,
the region of downflow, the perturbation of density is positive
and has a high value. When the density in a region exceeds a
threshold, the region is regarded as the boundary of a convective
cell. When a region is surrounded by one continuous boundary
region, the region is defined as one convective cell. Figure 7
shows the detected cells. Each color and each label (#n) correspond to each detected cell. We estimate the size of all cells and
compare the distribution of cell sizes for different ξ. The results
are shown in Fig. 8. The cell size distribution follows a power
law from about 0.01 to 10 Hr2 and there is no dependence on ξ in
the range from 1 to 20. Although when using this type of technique the size of cells tends to be large when neglecting smaller
cells, our conclusion is not wrong, since all auto-detections are
equally affected.
A30, page 5 of 7
A&A 539, A30 (2012)
0.06
Cell Size Distribution
106
Dist. Dens. (number H-2
r )
105
0.05
104
0.04
103
102
0.03
ξ=1
ξ=5
ξ=10
ξ=15
ξ=20
ξ=20/(δ/δr)1/2
101
100
10-1
0.001
0.010
0.100
1.000
Cell Size (H2r )
0.02
10.000
0.01
100.000
0.00
Fig. 8. Distribution of convective cell size with different ξ is shown. The
cell size distribution is averaged in time between t = 100 and 200Hr /vc .
(a) Mach number
1.00
0.10
0.01
(b) Vertical RMS velocity
1.0 1.5
Height (Hr)
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
2.0
(d) <-v⋅(∇ p1+ρ1g)>
0.015
0.010
0.005
0.005
0.000
0.000
-0.005
-0.005
-0.010
1.0
1.5
Height (Hr)
2.0
If the equation of continuity is satisfied, i.e. (∂ρ/∂t = −∇ · (ρu)),
the primitive form is obtained as
1.0 1.5
Height (Hr)
2.0
0.5
1.0 1.5
Height (Hr)
2.0
ξ=1
ξ=5
ξ=10
ξ=15
ξ=20
0.5
1.0
1.5
Height (Hr)
2.0
For the above two investigations, i.e. the Fourier analysis and
study of cell detection, we conclude that the statistical features
are unaffected by the RSST method as long as the effective Mach
number (computed with the reduced speed of sound) does not
exceed values of about 0.7, which corresponds to ξ = 20 in our
setup.
To confirm our criterion that the RSST is valid if the Mach
number is smaller than 0.7, we conduct calculations with larger
superadiabaticity, i.e. δr = 1 × 10−3 (cases 14–18). If our criterion is valid, required ξ with larger superadiabaticity must decrease. The results are shown in Fig. 9. Using δr = 1 × 10−3 ,
the calculations with ξ = 10, 15, and 20 generate a supersonic
convection flow near the surface (Fig. 9a). It is clear that the results for ξ = 10, 15, and 20 differ from those with ξ = 1 and 5.
The calculation with ξ = 5 corresponds to a flow whose Mach
number is around 0.6. Thus, our criterion is not violated even for
different values of the superadiabaticity.
Owing to our altered equation of continuity the primitive and
conservative formulation of Eqs. (3) to (5) are no longer equivalent. For example, the equation of motion in conservative form
is expressed as
∂
(ρu) = −∇ · (ρuu) + F,
(31)
∂t
where F denotes the pressure gradient, gravity, and Lorentz
force. With some transformations, we can obtain
∂u
∂ρ
u + ρ = −u∇ · (ρu) − ρ(u · ∇)u + F.
∂t
∂t
∂u
= ρ(u · ∇)u + F.
(33)
∂t
However, using the RSST these two forms are no longer equivalent. We used the primitive formulation at the expense that
energy and momentum are not strictly conserved; however, the
consistency of our results for different values of ξ strongly
indicates that this is not a serious problem for the setup we
considered. Alternatively, we could also implement our modified equation of continuity in a conservative formulation.
This would ensure that density, momentum, and energy are
strictly conserved at the expense of a modified set of primitive equations. Figure 10 shows the dependence of [∇ · (ρ0 u)]rms
(= [ξ2 ∂ρ1 /∂t]rms ) on ξ. Using ξ = 5, this term is almost the same
as that with ξ = 1. Although the deviation becomes large as ξ
increases, it is not proportional to ξ2 .
In the previous discussion, we kept ξ constant in the entire
computational domain. In the solar convection zone, the Mach
number varies however substantially with depth, from ∼1 in the
photosphere to <10−7 at the base of the convection zone, the
speed of sound itself varies from about 7 km s−1 in the photosphere to 200 km s−1 at the base of the convection zone. A reduction in the speed of sound is therefore most significant in the
deep convection zone, but not in the near surface layers. This
could be achieved with a depth dependent ξ. Even if we use a
conservative form of the equation of continuity,
1
∂ρ1
= −∇ · 2 ρ0 u ,
(34)
∂t
ξ
ρ
(e) <-ρ0v⋅(v⋅∇)v>
Fig. 9. The results with δr = 1 × 10−3 . The format is the same as Fig. 5.
A30, page 6 of 7
2.0
Fig. 10. Dependence of [∇ · (ρ0 u)]rms on ξ. In case 13, the value of ξ is
20 at the bottom and 4.5 at the top boundary.
-0.010
0.5
1.0
1.5
Height (Hr)
0.0
0.5
0.010
0.5
(c) Horizontal RMS velocity
0.6
0.0
0.5
0.015
ξ=1
ξ=5
ξ=10
ξ=15
ξ=20
ξ=20/(δ/δr)
(32)
the result must not be same as the result without any approximation of the statistical steady state. When the value averaged
during a statistical steady state is expressed as a, where a is a
physical value, the equation of continuity becomes
1
0 = ∇ · 2 ρ0 u ,
(35)
ξ
with inhomogeneous ξ. The solution of Eq. (35) is different from
the solution of the original equation of continuity in the statistical steady state, i.e. 0 = ∇ · (ρ0 u). Thus, the statistical
H. Hotta et al.: Convection with RSST (RN)
features such as rms velocity are not reproduced with an inhomogeneous ξ. If we use the non-conservative form of the equation of motion given in Eq. (2), this problem does not occur.
Thus, we investigate the nonuniformity of ξ in case 13 using the
non-conservative form, i.e., Eq. (3). To keep the Mach number
uniform for all heights, we use ξ = 20/(δ(z)/δr)1/2 , i.e. ξ = 20
at the bottom and ξ = 4.5 at the top boundary. We note that the
ratio of the speed
√ of convection to the speed of√sound is roughly
estimated as δ. In this setting, the value of δ is 1 × 10−2 at
the bottom and 4 × 10−2 at the top boundary. The same analysis
as that for uniform ξ is done (see Figs. 5, 6, and 10). There is
no significant difference between ξ = 20/(δ(z)/δr)1/2 and ξ = 1.
Although mass is not conserved locally for a non-homogeneous
ξ using a non-conservative form of the equation of continuity, the
horizontally averaged vertical mass flux is approximately zero
in the statistically steady convection and the conservation total
mass is not broken significantly. We conclude that an inhomogeneous ξ is valid under the previously obtained condition, i.e. the
Mach number is less than 0.7.
3. The base of the convection zone and the surface of the sun
where the anelastic approximation is broken can be connected, using a space-dependent ξ.
Overall we find that RSST is a very useful technique for studying
low Mach number flows in stellar convection zones as it substantially alleviates any stringent time-step constraints without
adding any computational overhead.
Acknowledgements. The authors want to thank Dr. M. Miesch for his helpful comments. Numerical computations were in part carried out on Cray XT4
at Center for Computational Astrophysics, CfCA, of National Astronomical
Observatory of Japan. We would like to acknowledge high-performance computing support provided by NCAR’s Computational and Information Systems
Laboratory, sponsored by the National Science Foundation. The National Center
for Astmospheric Reseach (NCAR) is sponsored by the National Science
Foundation. This work was supported by the JSPS Institutional Program for
Young Researcher Overseas Visits and the Research Fellowship from the JSPS
for Young Scientists. We have greatly benefited from the proofreading/editing assistance from the GCOE program. We thank the referee for helpful suggestions
that helped to improve our paper.
4. Summary and discussion
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We have applied RSST (described in Sect. 1) to 2D and 3D simulations of low Mach number thermal convection and confirmed
the validity of this approach as long as the effective Mach number (computed with the reduced speed of sound) stays below 0.7
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Since the Mach number is estimated to be 10−4 in the base of
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RSST and parallel computing with a large number of CPUs will
make it possible to calculate large-scale solar convection with
high resolution in the near future.
Compared to the anelastic approximation, there are three major advantages in RSST:
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code (regardless of the numerical scheme or grid structure)
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